Question

For which of the following quantities does ratios are dimensionless?
(i) `("work")/("Energy")` (ii) `sintheta` (iii) `(Momentum)/(Time)`

Answer 1

(i)Since work and energy both have the same dimensions `[ML^2T^(-2)]` , their ratio is a dimensionless quantity .
(ii)sin(i), Here I represents an angle.
An angle is the ratio of two lengths i.e., arc length and radius. Therefore I is dimensionless, hence sin(i) is dimensionless.
(iii)`["Momentum"/"Time"]=([MLT^(-1)])/"[T]"`
`=[MLT^(-2)]`
Hence, the given ratio is not dimensionless.